A random-phase approximation (RPA) treatment of edge magnetoplasmons (EMP) is presented for strong magnetic fields, low temperatures, and integer filling factors u. It is valid for negligible dissipation and lateral confining potentials smooth on the scale of the magnetic length ell_{0} but sufficiently steep that the Landau-level (LL) flattening can be neglected. LL coupling, screening by edge states, and nonlocal contributions to the current density are taken into account. In addition to the fundamental mode with typical dispersion relation omegasim q_x ln(q_{x}), fundamental modes with {it acoustic} dispersion relation omegasim q_x are obtained for u>2. For u=1,2 a {bf dipole} mode exists, with dispersion relation omegasim q_x^3, that is directly related to nonlocal responses.
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